HYBRID SOLAR PANEL EFFICIENCY OPTIMIZATION WITH A
LABYRINTH FIN ARRANGEMENT
by
Robert P. Collins
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December 11, 2013
i
© Copyright 2013
by
Robert P. Collins
All Rights Reserved
ii
ACKNOWLEDGMENT
Thank you father, mother, David, Matthew, and Alex for the example and inspiration
you provide in my life. I would not be who, or where I am today without your influence
on my life, and for that I am grateful.
Also, thank you Ernesto for the time and interest you took in my project as my advisor.
iii
CONTENTS
ACKNOWLEDGMENT .................................................................................................. iii
LIST OF TABLES ............................................................................................................ vi
LIST OF FIGURES ......................................................................................................... vii
TABLE OF SYMBOLS ................................................................................................. viii
KEYWORDS ..................................................................................................................... x
ABSTRACT ..................................................................................................................... xi
1. INTRODUCTION/BACKGROUND .......................................................................... 1
1.1
Solar Photovoltaic Cells ..................................................................................... 1
1.2
Solar Hot Water Heater ...................................................................................... 4
1.3
Hybrid Solar Panel (PV/T) ................................................................................. 5
2. METHODOLOGY/APPROACH ................................................................................ 7
2.1
Materials ............................................................................................................. 8
2.2
Model Arrangement ........................................................................................... 9
2.3
Test Arrangements ........................................................................................... 13
2.4
Model Theory and Relevant Equations ............................................................ 13
2.5
Finite Element Model ....................................................................................... 17
2.6
Expected Results .............................................................................................. 18
2.7
Model Limitations and Mesh Studies .............................................................. 18
3. RESULTS AND DISCUSSION ................................................................................ 20
3.1
PV/T Module Results ....................................................................................... 20
3.2
PV/T Array Results .......................................................................................... 30
3.3
Other Considerations ........................................................................................ 32
4. CONCLUSIONS ....................................................................................................... 33
REFERENCES ................................................................................................................ 34
APPENDIX A: CONSERVATION OF ENERGY CALCULATION ............................ 36
APPENDIX B: ELECTRICAL EFFICIENCY VERIFICATION .................................. 37
iv
APPENDIX C: THERMAL EFFICIENCY VERIFICATION ....................................... 38
APPENDIX D: VOLUME FLOW RATE ....................................................................... 39
APPENDIX E: TOTAL ENERGY COLLECTED ......................................................... 40
APPENDIX F: CONVECTIVE HEAT LOSS ................................................................ 41
APPENDIX G: RADIATIVE HEAT LOSS ................................................................... 42
v
LIST OF TABLES
Table 1: PV/T Model Materials ......................................................................................... 9
Table 2: Module Parameters ............................................................................................ 11
Table 3: Model Variables ................................................................................................ 12
Table 4: Fin Test Arrangements ...................................................................................... 13
Table 5: “Coarser” Mesh Solution Data and PC Specifications ...................................... 18
Table 6: Mesh Result Heat Balance and Accuracy ......................................................... 19
Table 7: Module Results .................................................................................................. 26
Table 8: Array Results ..................................................................................................... 32
vi
LIST OF FIGURES
Figure 1: Electrons Absorbing Incident Sunlight [1] ........................................................ 1
Figure 2: Band Gap [2] ...................................................................................................... 2
Figure 3: PV Array Hierarchy [6] ...................................................................................... 3
Figure 4: Active Secondary Loop Solar Hot Water Heater System [8] ............................ 4
Figure 5: Hybrid Solar Panel Control Volume .................................................................. 7
Figure 6: Model Isometric Cross Section View ................................................................ 8
Figure 9: Hybrid Panel Cross Section View .................................................................... 10
Figure 7: Fin Labyrinth .................................................................................................... 10
Figure 8: PV/T Module Landscape View ........................................................................ 10
Figure 10: Model Boundary Conditions .......................................................................... 12
Figure 11: Hydraulic Diameter ........................................................................................ 15
Figure 12: Rectangular Orifice [16] ................................................................................ 16
Figure 13: Averaged Streamlines and Contours of Turbulent Kinetic Energy [16] ........ 17
Figure 14: “Coarser” Mesh .............................................................................................. 17
Figure 15: Mesh Accuracy............................................................................................... 19
Figure 16: No Fin Velocity Profile .................................................................................. 21
Figure 17: Fin Velocity Disruption.................................................................................. 21
Figure 18: Velocity Distribution in a Labyrinth Arrangement ........................................ 22
Figure 19: No Fin Temperature Distribution ................................................................... 23
Figure 20: Temperature Distribution Around a Fin ......................................................... 23
Figure 21: Temperature Contours in a Labyrinth Arrangement ...................................... 24
Figure 22: PV Surface Temperature Distribution ............................................................ 25
Figure 23: PV/T Module Thermal and Electrical Efficiency Correlation ....................... 27
Figure 24: Top Fin Arrangement Efficiency ................................................................... 28
Figure 25: Bottom Fin Arrangement Efficiency .............................................................. 29
Figure 26: Labyrinth Arrangement Efficiency ................................................................ 29
Figure 27: All Fin Arrangements ..................................................................................... 30
Figure 28: PV/T Array [6] ............................................................................................... 31
vii
TABLE OF SYMBOLS
Symbol
𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
Description
Units
Collector Area Exposed to Solar Radia-
𝑚2
tion
𝐴𝑓𝑙𝑜𝑤
Flow Cross Sectional Area
𝐶𝑣
Constant Volume Specific Heat
𝐶𝑝
Constant Pressure Specific Heat
𝐷ℎ
Hydraulic Diameter
𝐺
Radiative Power Per Unit Area
h
Heat Transfer Coefficient
𝐻𝑓𝑙𝑜𝑤
𝑚2
𝐽
𝑘𝑔𝐾
𝐽
𝑘𝑔𝐾
𝑚
𝑊
𝑚2
𝑊
𝑚2 𝐾
Flow Path Height
m
𝐼
Current
𝐴
𝑘
Thermal Conductivity
𝑊
𝑚̇
Mass Flow Rate
𝑚𝐾
𝑘𝑔
𝑠
𝑃𝐸 𝑜𝑢𝑡
Electric Power Out of Panel
𝑊
𝑃𝑇 𝑜𝑢𝑡
Thermal Power Out of Panel
𝑊
𝑄̇𝑐𝑜𝑛𝑣
Energy Imparted on the Fluid
𝑊
𝑄̇𝑟𝑎𝑑
Energy Radiated to the Atmosphere
𝑊
Re
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
Reynolds Number
𝑇𝑎𝑚𝑏
Ambient Air Temperature
𝐾
𝑇𝑐𝑒𝑙𝑙
Cell Surface Temperature
K
𝑇𝑖
Inlet Working Fluid Temperature
𝐾
𝑇𝑜
Average Outlet Working Fluid
𝐾
Temperature
𝑇𝑟𝑜𝑜𝑚
u
Room Temperature
25 °C
Fluid Velocity at a Point
viii
𝑚
𝑠
Average Inlet Flow Velocity
𝑚
𝑉
Voltage
𝑉
w
Panel Width
m
PV Cell Temperature Coefficient
1
℃
𝑈𝑓𝑙𝑜𝑤
𝛽𝑟𝑒𝑓
𝑠
𝜀
Emissivity
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
𝜂𝐸
Panel Electrical Efficiency
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
Panel Electrical Efficiency at Room
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
𝜂𝐸𝑟𝑒𝑓
Temperature
𝜂𝑂
Panel Overall Efficiency
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
𝜂𝑇
Panel Thermal Efficiency
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
µ
Dynamic Viscosity
𝑘𝑔
𝑚∗𝑠
ν
𝑚2
Kinematic Viscosity
𝑠
ρ
Density
σ
The Stefan-Boltzmann Constant
∇
Gradient
𝑘𝑔
𝑚3
𝑊
5.67 × 10−8 𝑚−2 𝐾−4
𝑂𝑝𝑒𝑟𝑎𝑡𝑜𝑟
ix
KEYWORDS
Hybrid Solar Panel
Energy Conversion Efficiency
Heat Transfer
Low –Reynolds Number Turbulent Flow
Solar Energy
Finite Element Method
x
ABSTRACT
A photovoltaic (PV) cell is coupled with a solar hot water heater in an arrangement
called a hybrid solar panel, or PV/Thermal (PV/T) panel. This hybrid solar panel
concept explores the used of fins perpendicular to the flow direction to increase convection and reduce boundary layer thickness at a low Reynolds number in order to increase
heat transfer between the PV cells and solar hot water heater while achieving a useful
temperature rise. The hybrid panel is designed with solar cells attached to a copper
reservoir using a thermal paste, with an insulated boundary between the bottom of the
fluid reservoir and the atmosphere. A two dimensional (2-D) finite element model is
used to simulate the temperature distribution and the outlet water temperature in the
PV/T module, where the number of fins and the flow rate in the reservoir are varied.
The module efficiency is compared, with the highest efficiency module arrangement
consisting of many, large fins, in a labyrinth arrangement. The model of the highest
efficiency PV/T module is run three times, with the outlet water temperature carried
from one model to the next in order to simulate a larger, PV/T array, resulting in a water
temperature rise of 16.5°C, and an overall efficiency of 78.0%, 4.8% more efficient than
the PV/T array modeled with no fins.
xi
1. INTRODUCTION/BACKGROUND
The amount of electricity and hot water used in the world is increasing as the middle
class, and therefore the average quality of life, increases. This, coupled with the rise in
fossil fuel prices, creates a growing need for a cheap, energy efficient, and environmentally friendly method for providing electricity and hot water. Household solar hot water
heaters are a method of heating water, whereas photovoltaic (PV) cells are employed to
convert solar radiation into electricity. Both methods stated above provide alternatives
to fossil fuels by converting the sun’s light into usable energy, and are explored below.
1.1 Solar Photovoltaic Cells
The conversion of solar radiation into electrical power is accomplished in PV cells by
the use of semiconducting materials, with silicon as the most common semiconductor in
PV cells. During conversion of solar radiation to electrical power, photons of light are
absorbed by the valence electrons surrounding the nucleus of the silicon atoms. These
absorbed photons excite the electrons, raising them to a higher energy state, or to a
higher electron orbital Figure 1.
Figure 1: Electrons Absorbing Incident Sunlight [1]
The electron orbitals discussed are classified as valence band orbitals, the orbitals where
electrons are bound to an individual atom at resting state, and conduction band orbitals,
where electrons can move freely between different atoms in a material. The difference
in energy between the electron orbitals in the valence and conduction bands, where no
electron states can exist, is known as the band gap, and is a quantifiable number for a
1
given material and temperature. Materials with small or no band gap are classified as
conductors, whereas materials with a large band gap are classified as insulators. Materials such as silicon, which have an intermediate band gap, are semiconductors. Figure 2,
shows a schematic representation of the band structure of a semiconductor [2].
Figure 2: Band Gap [2]
This band gap is the step, or wall, that the electrons must overcome to move from the
valence band to the conduction band. In other words, the electrons need to be excited by
photons of a certain, minimum energy to jump the band gap. The band gap can be
considered proportional to the open circuit voltage of the semiconductor. With an
increase in temperature, the electrons have a higher resting energy state, effectively
reducing the band gap. With a reduction in band gap, the open circuit voltage of the
semiconductor of the PV cell decreases, while the current remains largely the same.
Because of Watt’s Law, P= 𝐼𝑉, the power output of the semiconductor, or PV cell
decreases for the same amount of power in from solar radiation. The electrical efficiency of a PV Cell is therefore decreased, as shown by the below equation [3], where the
current and voltage are measured at the maximum power point operation of the cell.
𝜂𝐸 =
𝑃𝐸 𝑜𝑢𝑡
𝐼∗𝑉
=
𝑃𝑖𝑛
𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
This phenomenon of a decrease in electrical efficiency with rising temperature is well
documented, with the relationship between temperature and efficiency explored by
%
Skoplaki, and Palyvos [4], which for some models, predicts a 0.41℃ drop in efficiency
2
above room temperature. In hot, high sunlight conditions, temperatures of 50°C can be
reached, severely dropping the efficiency of the panel, and also risking permanent
structural damage to the PV cell from the thermal stress [4].
𝜂𝐸 = 𝜂𝑇𝑟𝑒𝑓 (1 − 𝛽𝑟𝑒𝑓 (𝑇 − 𝑇𝑟𝑜𝑜𝑚 ))
The efficiency of the most common types of solar panels, mono-crystalline silicon PV
cells, typically ranges from 13-20% at room temperature [5], with that percentage of
power in sunlight converted into electrical power. The rest of the energy reflected off
the PV cells, or converted into heat. If that atmosphere is unable to accept the heat from
the PV cells, the temperature of the PV cells rises. As the temperature of the cells rise,
the efficiency of the cells decreases, and on hot, sunny days, PV cells can have a drop in
efficiency of up to 10%. Several methods are therefore used to cool PV cells in order to
maintain their electrical efficiency.
In practice, PV cells are arranged in modules, which are then combined to form PV
arrays; Figure 3 shows this hierarchy. This array is typical of the arrangement used for
the hybrid solar array used in this study.
PV Modules
PV Cells
Figure 3: PV Array Hierarchy [6]
3
1.2 Solar Hot Water Heater
A solar collector in a solar hot water heater is an enclosed volume that the working fluid
flows through to collect the sun’s energy in the form of heat. This volume is very
insulated and optimized to capture solar radiation. Solar hot water heaters have a greater
efficiency when the collector volume is hot, and there is a large driving temperature
delta between the collector and the working fluid flowing through the collector. The
working fluid is typically water in a single loop solar hot water heater, which directly
feeds water for household usage. Water and other fluids such as a water/propylene
glycol mix are used to transfer the heat to the household water supply through a secondary heat exchanger in a secondary loop solar hot water heater. The working fluid can be
supplied actively, with a pump, or passively using natural convection of the fluid as it is
heated. Passive, natural convection flow is typical of primary loop solar hot water
heaters, whereas active loops are used in both primary and secondary solar hot water
heaters. An active, secondary solar hot water heating arrangement, typical to the arrangement used in this study, is shown below in Figure 4. [7]
Figure 4: Active Secondary Loop Solar Hot Water Heater System [8]
4
The thermal efficiency of a solar hot water heater is given by the below equation [3]:
𝜂𝑇 =
𝑚̇ ∗ 𝐶𝑝 (𝑇𝑜 − 𝑇𝑖 )
𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
1.3 Hybrid Solar Panel (PV/T)
In order to avoid the PV cells drop in efficiency, and capture the majority of the waste
heat from the PV cells, the PV cells are coupled with a solar hot water heater in a hybrid
solar panel, or PV/T panel. This design provides a novel method for cooling the PV
cells, whose efficiency diminishes with increasing temperature, and uses the heat extracted from the PV cells to heat household or commercial hot water as an alternative to
traditional hot water heaters powered by fossil fuels. The efficiency of the hybrid solar
panel is the sum of the efficiency of the PV cells and the solar hot water heater [3].
𝜂𝑜 = 𝜂𝐸 + 𝜂𝑇
Hybrid solar panels, also known as PV/T panels, have been explored in several previous
studies. In a similar study Fountenault [9] varied flow rates and flow channel thicknesses in a laminar flow, hybrid solar panel. The exploration showed that, when all else was
constant, the average driving temperature difference between the PV cells and the fluid
in the flow channel drives both the thermal and electrical efficiencies. A large temperature delta was achieved by using a high mass flow rate of water in a large channel, which
led to lower temperature changes in the fluid when compared to lower flow rates in
smaller channels. As a result of a lower temperature change in the fluid, a larger average
temperature delta between the PV cells and the working fluid was maintained, and less
heat was lost to the atmosphere.
A study by Yang et al. [10] explored a model and prototype hybrid solar panel with a
functionally graded material (FGM). The FGM is a material with a property gradient.
𝑊
In this case, the thermal conductivity of the material is higher, 1.13𝑚𝐾, near the interface
5
𝑊
with the PV cells, and much lower, 0.26𝑚𝐾, near the bottom, freely convecting surface.
The FGM is intended transfer heat from the PV cells to the water, but also act as an
insulator between the water and the atmosphere, as the properties reflect. The study
showed that a combined efficiency of 70% could be achieved with hybrid solar panels,
and that there are several novel, although sometimes less practical, designs being explored to optimize hybrid solar panels as a means of harnessing the sun’s power.
6
2. METHODOLOGY/APPROACH
A hybrid solar panel is designed, with a set control volume that encapsulates the panel as
shown below in Figure 5. The sunlight and the cold working fluid will be the two
defined inputs into the control volume, and therefore solar panel. Heat will be transferred from the panel with the mass flow rate of the hot working fluid out, convectively
to the ambient air, and through reflection/radiation from the body of the body of the
panel. There is a current and voltage across the PV cell, which is also accounted for, and
a total electrical power out.
Radiation
Sunlight
Convective Heat
Transfer
Cold Primary Fluid
HYBRID SOLAR
Hot Primary Fluid
PANEL
(CONTROL VOLUME)
Figure 5: Hybrid Solar Panel Control Volume
The net energy balance for the hybrid solar panel control volume used in this study is:
𝐺 − 𝑃𝑇 𝑜𝑢𝑡 − 𝑃𝐸 𝑜𝑢𝑡 − 𝑄̇𝑐𝑜𝑛𝑣 − 𝑄̇𝑟𝑎𝑑 = 0
Within the control volume that is the hybrid solar panel, there is heat transfer between
the different material layers. Conduction heat transfer exists between and within the
solid layers of hybrid solar panel which will be constrained by the material conductivity.
Choice of highly conductive materials, such as copper will maximize the heat transfer
away from the solar panel to the walls of the cooling fluid reservoir. The heat transfer
out to the atmosphere is also minimized with a layer of insulation added to the bottom of
the hybrid solar panel.
7
Heat is conducted from the PV cells through highly conductive solids until it reaches the
solid/liquid boundary, where the heat is transferred to the working fluid in the fluid
reservoir. With known solid and liquid properties, the limiting factors explored in this
model are the surface area at the solid liquid boundary, boundary layers, and the convection in the flow path. Boundary layers form in duct flow, creating a hot layer of the
working fluid along the solid/liquid interface, where the bulk fluid temperature is much
lower. In a fluid such as water, conduction is a slower method of heat transfer than
convection. In order to increase the surface area, minimize boundary layer formation,
and increase heat transfer within the fluid by inducing mixing, or convection, sharp
edged fins are added perpendicular to the flow. A cross sectional unit thickness (not to
scale) of the model is shown below in Figure 6, which shows the material layers, and the
orientation of the fins to the flow.
- PV Cells
- Thermal Paste
- Copper
Working Fluid
- Copper
- Insulation
Figure 6: Model Isometric Cross Section View
2.1 Materials
Hybrid solar panel materials vary from the standard materials used by Fontenault [9] to
the FGM panel explored by Yang et al. Commonly available materials and those that
maximize heat transfer within the panel were chosen for this study. The materials used
in the model are listed below, with their reference and relevant material properties also
shown. COMSOL Multiphysics has built in materials, which are used for water, copper,
and silicon. All material properties below are constant in the model except for the
properties of water, which vary with temperature, with the water properties shown in
Table 1 are taken at 25°C.
8
Table 1: PV/T Model Materials
Material
PV Cell
Thermal Paste
Copper
Water
Property
Value
Reference
ρ
2329
COMSOL – Silicon
k
130
𝐶𝑝
700
ε
.60
[11]
ρ
3500
[12]
k
2.87
𝐶𝑝
.7
ρ
8700
k
400
𝐶𝑝
385
ρ
997.1
k
.611
𝐶𝑝 = 𝐶𝑣
4.184
μ
902 x 10-6
COMSOL – Copper
COMSOL – Water @ 25°C
2.2 Model Arrangement
A 2-D model of this scenario is created in COMSOL Multiphysics in order to simulate
this hybrid solar panel design. The number of fins on the top wall (0, 9, or 18), the
number of fins on the bottom of the wall (0, 9, or 18), and the fin length (1⁄4, 1⁄2, or
3⁄ flow path height) are tested for a single flow rate to see their effect on the efficiency
4
of the hybrid solar panel. The fins on the top of the flow path are expected to have a
two-fold effect on the heat transfer; increased flow mixing and increased surface area for
heat transfer on the “hot” wall. Fins on the bottom of the flow path do not increase the
surface area for heat transfer on the “hot” wall, but are instead tested for their ability to
disrupt boundary layer flow along the top wall and increase convection. Top and bottom
fins are also be tested together, as shown in Figure 7, which creates a labyrinth design,
which will effectively increase the flow path in the reservoir. For the labyrinth arrangement, an additional condition, 27 top and 27 bottom fins, will also be tested.
9
Figure 7: Fin Labyrinth
The PV/T module studied consists of 12 PV cells arranged in a 3x4 rectangle, with the
flow through the short direction of the rectangle, as shown in Figure 8. The model
orientation is shown in Figure 9, which illustrates a short cross section of the model.
Figure 8: PV/T Module Landscape View
Figure 9: Hybrid Panel Cross Section View
10
The 2-D model’s dimensions, inlets and initial conditions, and other non-material
properties are in Table 2. The PV cell properties and dimensions are taken for the
Suniva ARTisun Select [13], which is a standard size, high efficiency mono-crystalline
silicon PV cell. In areas with cold groundwater, such as Connecticut with an average
groundwater temperature of 11°C [14], cold water would enter the hybrid panel; however, water inlet temperatures lower than ambient would bias the model results, as the
surrounding, warmer air, would add heat to the model PV/T panel. As a result, an inlet
water temperature of 25°C, which matches the ambient temperature, is used in this
study.
Table 2: Module Parameters
Name
H_Flow
H_Wall
H_Paste
H_PV
H_Insulation
W_PVCell
H_Fin
W_Fin
U_Flow
T_Amb
T_Init
T_Inlet
Emissivity
HX_Silicon
W_Panel
PVEFF0
PVdeg
T_Room
Ltenth
Q_Sun
P_in
Expression
Unit
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[m/s]
[K]
[K]
[K]
Description
5
Flow Channel Thickness
2
Copper Wall Thickness
0.5
Thermal Paste Thickness
0.2
PV Cell Thickness
10
Insulation Thickness
125
Cell Width
1/2*H_Flow
Fin Height
1
Fin Width
0.002
Flow Inlet Velocity
298.15
Ambient Air Temperature
298.15
Initial Cell Temperature
298.15
Inlet Water Temperature
0.6
Emissivity Silicon
10.52 [W/(m^2*K)] Silicon/Air Heat Transfer Coefficient
3*W_PVCell
Width of Three PV Cells
0.182
PV Cell Efficiency at Room Temperature
0.0041 [1/K]
PV Cell Degradation With Temperature
298.15
Room Temperature
W_Panel/9.5
Fin Spacing (9 fins)
1000 [W/m^2]
Sun Incident Radiation
Q_Sun*W_Panel*1[m]
Power In
The heat transfer coefficient for the top of the panel is 10.52
𝑊
𝑚2 𝐾
[5], which assumes a
wind speed of approximately 1 m/s. This wind speed assumption is consistent with
11
Fontenault [9] and Yang et al. [10]. In a typical arrangement, the bottom boundary of a
hybrid solar panel is insulated with materials such as extruded polystyrene, which
reduces convective heat loss to the environment. Including insulation in the model
increases the number of elements that add little value to the study’s results. In lieu of
adding insulation to the model, an insulated boundary condition is used for the bottom
surface of the PV/T module. These boundary conditions are illustrated in Figure 10.
Figure 10: Model Boundary Conditions
The model variables in Table 3 are evaluated by the model for each step in the solver,
and are used iteratively to find the steady state solution for the model. An important
variable to note is Q_heat, which is the sun’s radiation that is not converted to electrical
power by the solar panel, which varies with temperature.
Table 3: Model Variables
Name
PVEFF
Q_Heat
mdot
ThermEFF
EFF_Net
Expression
PVEFF0*(1-PVdeg*(T-T_Room))
Q_Sun*(1-PVEFF)
nitf.rho*H_Flow*U_Flow*1[m]
mdot*nitf.Cp*(T-T_Inlet)/P_in
PVEFF+ThermEFF
Units
W
kg/s
-
12
Description
PV Cell Efficiency Temp. Dependence
Sun’s Energy Converted to Heat
Mass Flow Rate Water (per unit depth)
Thermal Efficiency
Overall Efficiency
2.3 Test Arrangements
A 2-D, stationary COMSOL model is used to study the hybrid solar panel. Multiple
avenues of heat transfer within, and in and out of the control volume are explored in this
model. Also, the fins introduced into the flow path will create localized flow separation
in the low Reynolds number flow. As a result of the conditions tested, a low Reynolds
number turbulent flow, conjugate heat transfer model is used. The fin test arrangements
are shown below in Table 4.
Table 4: Fin Test Arrangements
Flow Velocity (u)
# Top Fins
# Bottom Fins
Fin Lengths
.002 m/s
0
0
0
002 m/s
9
0
¼, ½, ¾
002 m/s
18
0
¼, ½, ¾
002 m/s
0
9
¼, ½, ¾
002 m/s
0
18
¼, ½, ¾
002 m/s
9
9
¼, ½, ¾
002 m/s
9
9
¼, ½, ¾
002 m/s
18
18
¼, ½, ¾
002 m/s
27
27
¼, ½, ¾
Twenty five fin arrangements are outlined in Table 4. These conditions are expected to
show a useful correlation between the different fin arrangements and the outlet thermal
properties and efficiencies of a PV/T module.
2.4 Model Theory and Relevant Equations
The relevant equations used in this model and for the purpose of determining the panel’s
thermal properties overall efficiency are discussed in this section. The thermal, electrical, and overall efficiency equations of the hybrid panel are mentioned above in the
introduction. The relevant equations in the COMSOL conjugate heat transfer model are
13
shown below. A simplified computation was used to verify the model’s relevance for
modeling the conditions, with vector quantities shown in bold.
Steady state heat transfer in solids with no heat generation is described by the conservation of energy equation
∇ ∙ (𝑘∇𝑇) = 0
where k is the thermal conductivity of the solid.
Steady state heat transfer in liquids is also described by the conservation of energy
equation, where 𝜌𝐶𝑝 𝒖 ∙ ∇𝑇 is the rate of convective heat transfer in the fluid.
𝜌𝐶𝑝 𝒖 ∙ ∇𝑇 = ∇ ∙ (𝑘∇𝑇)
The heat flux from solar irradiance into the PV cell is given by
−𝐧 ∙ (−𝑘∇𝑇) = 𝐺
Where 𝐺 is a defined value of the incident heat flux and 𝐧 is the vector normal to the
heat transfer surface. This equation is also used to describe a perfectly insulated boundary. Thermal Insulation in the model means there is no heat transfer across a given
boundary, which essentially means the temperature gradient leading up to and across the
boundary is zero.
−𝐧 ∙ (−𝑘∇𝑇) = 0
Free convection between the atmosphere and the hybrid solar panel is based on the heat
transfer coefficient and the temperature difference between the atmosphere and the
surface of the panel. The convective heat loss from the panel to the atmosphere is given
by the below equation where h is the heat transfer coefficient.
14
−𝐧 ∙ (−𝑘∇𝑇) = ℎ ∙ (𝑇𝑎𝑚𝑏 − 𝑇)
Radiative heat transfer is in included in the model, and is found to be considerable when
the PV/T array is heated several degrees above ambient.
−𝐧 ∙ (−𝑘∇𝑇) = 𝜀σ(𝑇𝑎𝑚𝑏 4 − 𝑇 4 )
A no slip boundary condition is used at the solid/fluid interface, with the fluid velocity
set as zero along the walls of the flow path. The velocity profile of the fluid is given by:
𝐮 = −𝑈𝑓𝑙𝑜𝑤 𝒏
where 𝑈𝑓𝑙𝑜𝑤 is the initial average velocity, which is a defined test condition. The
Reynolds number is evaluated at the tip of each fin to evaluate mixing, with the hydraulic diameter, 𝐷ℎ , as twice the flow path. Figure 11 shows a parabolic velocity profile,
typical to laminar flow, and the flow path height.
𝑅𝑒 =
𝑢𝐷ℎ
𝜈
𝐷ℎ = 2𝐻𝑓𝑙𝑜𝑤
𝑢
𝐻𝑓𝑙𝑜𝑤
Figure 11: Hydraulic Diameter
For flow between two parallel plates with the model geometry in Table 2, the Reynolds
number is calculated below.
15
𝑅𝑒 =
𝑚
𝑠
.005𝑚
))
2
2
𝑚
1.307𝑥10−6
𝑠
(.002 )(4(
= 15.3
The flow through the “no-fin” model has a Reynolds number well below that of turbulent flow, with the transition between laminar and turbulent flow occurring at a Reynolds
number between 2300 and 4000 [15].
The mixing in the model is therefore accounted for by low-Reynolds number turbulent
flow. This behavior is typical of flow through an orifice or diffuser. Diffuser stall,
which is a term typically used in aerofoil aerodynamics, denotes boundary layer separation and is explained by White: “The expanding-area diffuser produces low velocity and
increasing pressure, an adverse gradient. It the diffuser angle is too large, the adverse
gradient is excessive, and the boundary layer will separate at one or both walls, with
backflow, and poor pressure recovery” [15]. It is this boundary layer separation or
disruption which is relied upon to increase fluid mixing and therefore heat transfer in the
model. Figure 12, below, shows the rectangular orifice modeled by Tsukahara, Kawase,
and Kawaguchi [16], with the turbulent kinetic energy of a Newtonian fluid shown in
Figure 13. The simulation was carried out with a Reynolds number of 100, and shows
that the reduction in area through the orifice disrupts the normal laminar flow boundary
layers and introduces turbulent kinetic energy in the form of flow mixing.
Figure 12: Rectangular Orifice [16]
16
Figure 13: Averaged Streamlines and Contours of Turbulent Kinetic Energy [16]
Figure 12 shows that the flow has increased energy as a result of the expanding area,
causing enhanced mixing, which is the behavior that is expected in this model. Similar
to Tsukahara, Kawase, and Kawaguchi [16], sharp edged bodies, or fins, are used, which
are insensitive to Reynolds number and “cause flow separation regardless of the character of the boundary layer” [15].
2.5 Finite Element Model
The hybrid solar panel model is meshed using the “Coarser” Physics Controlled Mesh in
COMSOL Multiphysics. COMSOL uses a segregated solver, with two groups that
converge to a single solution, for the low Reynolds number turbulent flow k-ε. The
segregated solver is computationally complex, and even for a “Coarser mesh”, approximately 50,000 elements are created for the more simple models with fewer fins.
Solutions with more fins are more computationally demanding, but the “Coarser” mesh
was still used to maintain the integrity of the results. An example mesh is shown below
in Figure 14, with the finite element mesh data shown in Table 5.
Figure 14: “Coarser” Mesh
17
Table 5: “Coarser” Mesh Solution Data and PC Specifications
Objects, Domains, Boundaries, Vertices
42
42
199
158
Elements
Domain: 64,444
Boundary: 5,989
Degrees of Freedom
Solver 1: 27,804
Solver 2: 177,715
Solution Time
6 minutes, 18 seconds
PC Type
Lenovo PG101
PC Processor
Intel i3 – 3220; 3.30 GHz
PC RAM
4 GB
Studies involving 27 fins on the top and bottom of the flow path require the use of, an
“Extra Coarse” mesh, because the segregated solver runs out of memory during its
Lower/Upper matrix factorization for a “Coarser” mesh. With the “Coarser” mesh, the
number of degrees of freedom approached 250,000 for the second solver.
2.6 Expected Results
The expectation is that heat transfer will be maximized when the surface area in contact
with the fluid is maximized with the top fins, the boundary layer and the extent of the
dead flow zones in front and behind the fins is interrupted by the bottom fins, and the
flow path is extended with the labyrinth arrangement. In summary, the expectation is
that the greatest heat transfer will occur with the largest number of large fins on the top
and bottom of the flow path. Altogether, the efficiency of both entities in the panel, the
PV cells and the solar hot water heater, are expected to reach their peak when heat
transfer between the two components of the hybrid solar panel are maximized.
2.7 Model Limitations and Mesh Studies
A general proof of concept calculation, in Appendix A shows that the model, without
heat transfer to the surrounding atmosphere, is about 90% accurate. That is to say that
10% of the heat incident on the module is unaccounted for in the single heat outlet; the
“hot” water. These limitations are especially noticeable under the low flow, small flow
height conditions, which this model is simulating. Higher flow rates, or larger flow
18
heights were not used, because they would not yield the useful temperature delta that is a
design requirement for a hybrid solar panel.
The accuracy of the results is also dependent on the mesh, with a finer mesh yielding
more accurate results. The accuracy and error of the model with no fins, insulated
boundaries, and 375 W of inlet heat transfer is shown below in Table 6 and then plotted
in Figure 15. The inlet heat in the model with insulated boundaries can only be transferred to the fluid. The added energy is the difference between the outlet fluid energy,
Qfluid Out, and the inlet fluid energy, Qfluid In. The computer used to run the model,
specifications in Table 5, ran out of memory for all meshes finer than the “coarse” mesh.
More accurate results could be achieved with a computer with more processing power.
Table 6: Mesh Result Heat Balance and Accuracy
Mesh
Extremely
Coarse
Extra
Coarse
Coarser
Coarse
Chart
Point
1
Qfluid In
(J)
Qfluid Out
(J)
Delta Q
(J)
Qheat In
(J)
% Error
% Accuracy
11880
12072
192
375
48.8
51.2
11889
11892
11895
12181
12227
12240
292
335
345
375
375
375
22.1
10.7
8.0
77.9
89.3
92.0
2
3
4
Mesh Accurary
100.0
% Accuracy
80.0
60.0
40.0
20.0
0.0
1
2
3
Mesh
Figure 15: Mesh Accuracy
19
4
3. RESULTS AND DISCUSSION
3.1 PV/T Module Results
The 2-D model is run for the PV/T module with the flow length of three PV Cells for
each of the test arrangements outlined in Table 4. The parameters used in the model are
outlined in Table 2, which represent typical hybrid solar panel dimensions, properties,
and typical test operating conditions. COMSOL Multiphysics iteratively solves the
finite element mesh of the PV/T module using the equations stated in Section 3.4. Outlet
water temperatures and PV cell surface temperatures are averaged by COMSOL for each
test arrangement. COMSOL also uses the equations for the variables shown in Table 3
to calculate the electrical, thermal, and overall efficiency of the PV/T module; 𝜂𝑒 , 𝜂𝑡 ,
and 𝜂𝑜 respectively. The electrical and thermal efficiency values are also calculated by
hand in Appendixes B and C using the inputs from Table 2 the COMSOL output temperatures. The calculation of the electrical efficiency differed only slightly from the
COMSOL model result; the hand calculation in Appendix B uses the averaged cell
temperature to calculate the efficiency, whereas the model calculates the efficiency at
each element of the cell and then averages, which is a more accurate method. Rounding
error explains the minor difference between the model and Appendix C results of the
thermal efficiency, as both use the averaged outlet water temperature calculated by
COMSOL. The overall efficiency is calculated by addition and is visually verified.
The COMSOL 2-D model result for the velocity distribution for the case with no fins is
shown in Figure 16.
This parabolic velocity profile, represents a fully developed,
laminar flow profile, where there are layers of fluid parallel to the flow path, with little
mixing.
Flow around fins perpendicular to the flow path, which are added to induce
low Reynolds number turbulent flow, is shown in Figure 17. The no slip boundary
condition is noted along the walls of the flow path, where the velocity is zero at the
walls. The fins introduced into the flow disrupt the developed flow profile exhibited in
between fins. Water is accelerated as the flow path height decreases at the tips of the
fins, and the flow begins to decelerate as the area suddenly increases after each fin. As
20
the flow decelerates, the energy kinetic energy is converted to pressure, creating an
adverse gradient, flow separation, and therefore increased convection [15]. The use of
sharp edged fins assures separation despite the low bulk Reynolds number of the fluid.
Figure 16: No Fin Velocity Profile
Figure 17: Fin Velocity Disruption
21
It should be noted from Figure 17 that the flow quickly re-establishes itself after travelling past a fin. Figure 18 illustrates that, as the fins are moved closer together, the flow
spends less time at a constant velocity, and is instead constantly accelerated across each
fin and decelerated in between the fins. The flow path length is also increased as more
fins are added and the spacing between fins decreases. Flow no longer travels directly
across the panel, but instead crisscrosses between a labyrinth of fins. This arrangement
increases distance the flow travels, without increasing the length of the PV/T module.
Figure 18: Velocity Distribution in a Labyrinth Arrangement
Heat is transferred from the PV cells, through the highly conductive thermal paste and
copper wall, and into the fluid reservoir. The temperature distribution arrangement with
no fins is shown in Figure 19, with lines of constant temperature slowly disappearing
along the flow path. Lines of constant temperature around a fin are illustrated in Figure
20. The lines of constant temperature contour around the fin, illustrating that the fin is a
heat source to the fluid. The fins add surface area to the “hot” top wall, which increases
the heat transferred to the fluid, raising the water temperature and cooling the PV cells.
22
Figure 19: No Fin Temperature Distribution
Figure 20: Temperature Distribution Around a Fin
23
Lines of constant temperature are shown in Figure 21 below for a labyrinth fin arrangement. The flow is mixing, causing the fluid temperature to be more evenly distributed.
In Figures 19 and 20, there are many lines of constant temperature that show layers of
water with different temperatures. By contrast, Figure 21 shows a flow with a more
evenly distributed temperature, with fewer lines of constant temperature and the lines
disappearing as the flow mixes and weaves through the labyrinth of fins.
Figure 21: Temperature Contours in a Labyrinth Arrangement
The panel surface temperature has a non-linear distribution as shown in Figure 22. Heat
transfer between the hot solid layers at the top of the PV/T module and the cooling flow
is directly proportional to the driving temperature difference between the two. The
water temperature increases as it travels through the PV/T module, and, with a smaller
temperature difference between the panel and the cooling fluid, there is less heat transfer
to the fluid. The thermal profile in Figure 22 shows that, heat is transferred from the hot
fluid outlet end of the panel through the highly conductive PV, thermal paste, and copper
layers to the cold fluid inlet end. As a result, there is a greater amount of heat transfer
24
and therefore a larger temperature rise at the cold fluid inlet end of the panel, and there is
a downward concavity to the curve in Figure 22.
Figure 22: PV Surface Temperature Distribution
The results of each model run are shown in Table 7, related to their fin arrangements
proposed in Table 4. The water inlet temperature for each condition listed below is 25°C
(298.15 K), with an average inlet velocity of .002 m/s and flow path height of 5mm. The
other parameters that remain constant are listed in Table 2.
25
Table 7: Module Results
Number
Fin Length*
𝑇𝑜 (K)
𝑇 𝐶𝑒𝑙𝑙𝑎𝑣𝑔 (K)
𝜂𝐸
𝜂𝑇
𝜂𝑜
None
0
0
303.75
302.99
17.84
62.11
79.95
9
¼
303.77
302.97
17.84
62.31
80.15
9
½
303.80
302.92
17.84
62.68
80.52
9
¾
303.88
302.78
17.85
63.53
81.38
18
¼
303.77
302.96
17.84
62.32
80.16
18
½
303.83
302.87
17.85
62.96
80.81
18
¾
303.97
302.67
17.86
64.57
82.43
9
¼
303.77
302.95
17.84
62.35
80.19
9
½
303.80
302.86
17.85
62.63
80.48
9
¾
303.85
302.71
17.86
63.24
81.10
18
¼
303.78
302.91
17.85
62.37
80.22
18
½
303.83
302.74
17.86
62.94
80.80
18
¾
303.92
302.49
17.88
63.94
81.82
9
¼
303.77
302.94
17.84
62.36
80.20
9
½
303.83
302.81
17.85
62.93
80.78
9
¾
303.94
302.60
17.87
64.24
82.11
18
¼
303.80
302.88
17.85
62.66
80.51
18
½
303.90
302.67
17.86
63.77
81.63
18
¾
304.06
302.37
17.89
65.52
83.41
27
¼
303.84
302.81
17.85
63.07
80.92
27
½
303.98
302.54
17.87
64.67
82.54
27
¾
304.25
302.31
17.89
67.67
85.65
(Top and Bottom)
Labyrinth
Bottom
Top
Fins
The electrical efficiency of the PV cell varied only slightly with each case, with a lowest
efficiency of 17.84% at the no-fin condition and only 17.89% for the most efficient,
many, large fin labyrinth condition. This is behavior is caused by the large driving
temperature difference between the PV Cells and the cooling water as well as the already
low thermal resistance between the two. Adding fins perpendicular to the flow path has
a slightly more dramatic effect on the thermal efficiency of the hybrid solar panel, as the
flow separation does cause the fluid to mix more, and therefore accept more energy from
26
to PV cell. The thermal efficiency varies between 62.1% and 67.7%, a 5.6% difference
between the arrangement thermal efficiencies. The thermal efficiency is correlated with
electrical efficiency in Figure 23 below.
PV/T Module Efficiency Correlation
68.00
Thermal Efficiency
67.00
66.00
65.00
64.00
63.00
62.00
61.00
17.83
17.84
17.85
17.86
17.87
17.88
17.89
17.90
Electrical Efficiency
Figure 23: PV/T Module Thermal and Electrical Efficiency Correlation
As expected, the electrical and mechanical efficiencies are correlated. As the thermal
efficiency increases, more heat is transferred away from the PV cells, keeping the cells
at a lower operating temperature. Consistent with semiconductor properties, PV cell
efficiency, or hybrid solar panel electrical efficiency, is inversely related to the operating
temperature. Due to the small changes in electrical efficiency, net or overall efficiency
of the PV/T module is largely governed by the thermal efficiency.
Efficiency of the PV/T module is dependent on the fin length for fins both on the top and
the bottom of the flow channel. Longer fins perpendicular to the flow path increase the
efficiency of the hybrid solar panels when compared to small fins. The large fins create
the largest flow disruption, mixing the fluid. The results for the top and bottom fin
arrangements are shown in Figures 24 and 25 respectively.
27
Top Fin Arrangement vs. Overall Efficiency
86
Overall Efficiency
85
84
83
9 Top Fins
82
18 Top Fins
81
80
79
1/4
1/2
3/4
Fin Length
Figure 24: Top Fin Arrangement Efficiency
The top fins are shown to have a more dramatic effect than bottom fins, as the top fins
increase surface area of the “hot” top wall while still causing separation of boundary
layers in the flow. By contrast, the bottom fins only contribute to flow mixing and
disruption of boundary layers. This is evident when comparing the efficiency graphs of
the top vs. bottom fins; Figures 24 and 25 respectively.
As expected, top and bottom fins together yield the highest efficiencies for the same
number of fins. Flow is mixed due to the addition of top and bottom fins, the surface
area of the “hot” boundary is increased with the addition of top fins, and the flow path
length is increased as the flow has to “crisscross” over top and bottom fins as illustrated
in Figure 18. The overall efficiencies of the labyrinth arrangement results are shown in
Figure 26.
28
Bottom Fin Arrangement vs. Overall Efficiency
86
Overall Efficiency
85
84
83
9 Bottom Fins
82
18 Bottom Fins
81
80
79
1/4
1/2
3/4
Fin Length
Figure 25: Bottom Fin Arrangement Efficiency
Labyrinth Fin Arrangement vs. Overall Efficiency
86
Overall Efficiency
85
84
83
9 Labyrinth
82
18 Labyrinth
81
27 Labyrinth
80
79
1/4
1/2
3/4
Fin Length
Figure 26: Labyrinth Arrangement Efficiency
29
The highest overall efficiency is achieved for the arrangement with the largest number of
fins with fins on the top and bottom of the flow path and the largest fin size. This is
evidenced in Figure 26, which shows the case with 27 fins on the top and bottom of the
flow path and fins ¾ as long as the flow path height. Figure 27 shows a comparison of
all arrangements together for a full comparison, where the labyrinth flow arrangement
with 27 fins is clearly the most efficient arrangement for a given fin length.
All Fin Arrangements vs. Overall Efficiency
86
Overall Efficiency
85
27 Labyrinth
84
18 Labyrinth
83
18 Top Fins
82
9 Labyrinth
81
18 Bottom Fins
80
9 Bottom Fins
9 Top Fins
79
1/4
1/2
3/4
Fin Length
Figure 27: All Fin Arrangements
3.2 PV/T Array Results
With the PV/T module results evaluated, the most efficient module case is repeated in an
array. From section 4.2, the most efficient module was the case with many, large fins in
a labyrinth arrangement. This is compared to an array with no fins for an overall comparison to the outlet water temperature and efficiency. The array is shown in Figure 28,
which consists of three modules, Figure 8, linked together. The boundary between each
module is assumed to be perfectly insulated, with only the outlet water temperature
carried from one module to the next. A perfectly insulated boundary assumption allows
30
for each module to be run as a separate entity, removing concerns of conduction from
the outlet, hot end of the array affecting the results already calculated at the cold, inlet
end. Also, the insulated boundary condition better reflects the use of a gasket material
between modules to prevent leaks in the flow path.
Figure 28: PV/T Array [6]
The array results are displayed in Table 8, with the same inlet and boundary conditions
as the PV/T module, including the inlet water temperature of 298.15 K. As an array, the
𝐿
hybrid solar panel with a labyrinth fin arrangement delivers 0.3 𝑚𝑖𝑛, calculated in
Appendix D, of water that has been heated by 16.5 degrees Celsius. Without fins, the
model array heats the water to a temperature of 15.06 degrees, 1.4 degrees less than the
arrangement with fins. The energy transferred to the water has also cooled the PV cells,
maintaining, in both cases a similar cell operating efficiency. There is more heat transferred to the water in the case with the labyrinth fin arrangement, creating a higher
thermal efficiency; however, as the water temperature rises, the PV cell temperature
increases, decreasing the PV cell efficiency towards the end of the array. As a result of
this, the average electrical efficiency of the arrays, with and without fins, is about equal
at 17.5%, despite the higher electrical of the first module with the labyrinth fin arrangement.
31
Table 8: Array Results
Fins
Number
Fin Length*
𝑇𝑜 (K)
𝑇 𝐶𝑒𝑙𝑙𝑎𝑣𝑔 (K)
𝜂𝑒
𝜂𝑡
𝜂𝑜
None
0
0
313.21
307.88
17.47
55.68
73.15
Labyrinth
27
¾
314.66
307.79
17.48
60.93
78.02
The difference in overall efficiency is therefore controlled by the thermal efficiency, as
noted in Table 8. The amount of energy recouped from the environment is calculated in
the Appendix E, is a total of 440 W, for the conditions listed in Table 2.
As designed, the water and cell temperature of the hybrid solar panel continually rise as
flow travels along the flow path, so the heat transfer due to convection and radiation
increase. There is a greater amount of heat lost to the atmosphere and, as result, the
thermal efficiency of the PV/T array is lower than the module efficiency for both the
arrangement with no fins and the labyrinth. The convective and radiative heat losses to
the atmosphere for the array are calculated in Appendixes F and G respectively. The
values are 𝑄̇𝑐𝑜𝑛𝑣 = −57.0 𝑊 and 𝑄̇𝑟𝑎𝑑 = −20.5 𝑊, showing the heat losses are considerable as the array reaches high temperatures.
3.3 Other Considerations
Because of this model’s 2-D nature, heat exchange structures such as pins were not
explored. An arrangement of many, small, cylindrical pins are expected to have a
positive effect on heat transfer between the fluid and the PV Cell. For a very space or
weight limited application, where cost doesn’t have as much of an impact, a porous
media heat exchange process might also be explored.
32
4. CONCLUSIONS
The greatest overall PV/T module efficiency of 85.7% occurs with the labyrinth arrangement or the arrangement with 27 top and 27 bottom fins that are ¾ the height of the
flow path. This is an approximate 5.7% increase in efficiency over the arrangement with
no fins. Sharp edged fins are used to cause flow separation, which mixes the fluid
despite the low Reynolds number and regardless of the boundary layer formation. Not
only is the flow mixing increased, but the flow path has been extended, as the flow
crisscrosses around the fins at the top and the bottom of the flow path. This increases
surface area between the working fluid and the “hot”, upper heat transfer boundary,
without increasing the length of the hybrid solar panel. The thermal efficiency has the
greatest variation, with the PV cell efficiency kept relatively constant due to the small
temperature differences of the PV cell temperature between each arrangement.
When connected as an array, three modules linked in a head to tail arrangement, heat the
water by 16.5 degrees Celsius, collecting 440 W from the environment in the form of
usable electrical and thermal energy. The fins in the array provide a 4.8% increase in the
overall efficiency over the array without fins, 78.0% vs. 73.2% respectively. Despite the
model limitations, fins perpendicular to the hybrid solar panel flow path are shown in
this model to increase the heat transfer between the PV cells and the cooling water,
increasing the amount of energy collected from the sun’s light.
As predicted, the efficiency of a hybrid solar panel can be increased with fins perpendicular to the flow path. The efficiency increase is dependent on the number, size, and
arrangement of the fins, with the ideal arrangement consisting of many, large fins,
alternating between the top and bottom of the flow path in the direction of flow. Methods such as the labyrinth flow arrangement in a hybrid solar panel can be used to
optimize the energy conversion efficiency from solar energy, to a useful alternative to
fossil fuels.
33
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16. Tsukahara, Takahiro, Kawase, Tomohiro, and Kawaguchi, Yasuo. DNS of Viscoelastic Turbulent Channel Flow with Rectangular Orifice at Low Reynolds
Number. International Journal of Heat and Fluid Flow Volume 32, Issue 2011,
Pages 529-538.
35
APPENDIX A: CONSERVATION OF ENERGY CALCULATION
For a control volume, using the conservation of energy, the first law of thermodynamics
with no heat generation is
𝑄𝑖 = 𝑄𝑜𝑢𝑡
The inlet energy is the energy incident from the sun, taken for a unit length to relate to
the temperature rise in the 2-D model
𝑄𝑖 = 𝑞 ′′ 𝑠𝑢𝑛 𝐴𝐶𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 = 1000
𝑊
(. 375𝑚)(1𝑚) = 375 𝑊
𝑚2
The outlet energy is the energy absorbed by the fluid
𝑄𝑜𝑢𝑡 = 𝑚̇𝐶𝑝 ∆𝑇
Per unit width, the mass flow rate is the flow is
𝑚̇ = 𝜌 𝑢𝑓𝑙𝑜𝑤 𝐴𝑓𝑙𝑜𝑤 = 𝜌 𝑢𝑓𝑙𝑜𝑤 𝐻𝑓𝑙𝑜𝑤 1 𝑚
𝑚̇ = 1000
𝑘𝑔
𝑚
𝑘𝑔
∗
.002
∗
.005𝑚
∗
1𝑚
=
.01
𝑚3
𝑠
𝑠
The temperature difference is then solved by setting the energy inlet equal to the energy
added to the flow.
𝑄𝑜𝑢𝑡 = 𝑄𝑖 = 375 𝑊 = (. 01
𝑘𝑔
𝑊
) (4175
) (∆𝑇)
𝑠
𝑚𝐾
∆𝑇 = 8.98 𝐾
∆𝑇𝑚𝑜𝑑𝑒𝑙 = 8.07 𝐾
36
APPENDIX B: ELECTRICAL EFFICIENCY VERIFICATION
The equation for the electrical efficiency of a hybrid solar panel is dependent on the PV
cell temperature. The COMSOL model evaluates the efficiency at each element of the
PV cell layer in the model and then determines an average efficiency. The simple
calculation to verify the electrical efficiency is accomplished using only the average cell
temperature of the labyrinth arrangement with 27 top and bottom fins of ¾ flow path
height and the below equation [3].
𝜂𝐸 = 𝜂𝑇𝑟𝑒𝑓 (1 − 𝛽𝑟𝑒𝑓 (𝑇 − 𝑇𝑟𝑜𝑜𝑚 ))
The values of the room temperature efficiency, 𝜂𝑇𝑟𝑒𝑓 , the temperature coefficient of
mono-crystalline silicon cells, 𝛽𝑟𝑒𝑓 , and the room temperature, 𝑇𝑟𝑜𝑜𝑚 are taken from
Table 2, with the value for the average cell temperature in Table 4.
𝜂𝐸 = .182(1 − .0041
1
(302.31K − 298.15𝐾)) = 17.89%
℃
The a temperature difference is the same in Celsius as Kelvin, with the temperature
values left in Kelvin for convenience.
This is the same value calculated by the
COMSOL model for each cell element and then averaged. Using the average temperature of the PV cells is a very good estimate for the small temperature differences and
relatively linear temperature profile of the model results
37
APPENDIX C: THERMAL EFFICIENCY VERIFICATION
The thermal efficiency of the labyrinth arrangement with 27 top and bottom fins of ¾
flow path height is calculated using the below equation [3].
𝜂𝑇 =
𝑚̇ ∗ 𝐶𝑝 (𝑇𝑜 − 𝑇𝑖 )
𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟
The output water temperature value from the PV/T module and inputs for the mass flow
rate, 𝑚̇, and the sun’s inlet radiative power in, 𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 , previously calculated in
Section 7.1, and 𝐺 ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 , or the sun’s power into the panel, is 375 𝑊 are used to
calculate the efficiency. The specific heat value is taken at the outlet water temperature
of approximately 15°C.
𝑘𝑔
𝐽
(304.25K − 298.15𝐾)
. 01 𝑠 ∗ 4179
𝑘𝑔𝐾
𝜂𝑇 =
= 67.98%
375 𝑊
This value closely matches the 67.67% calculated by the model, and can be explained by
rounding error in this hand calculation.
38
APPENDIX D: VOLUME FLOW RATE
The cross sectional area of the flow path is the height of the flow path multiplied by the
panel, or array width.
𝐴𝑓𝑙𝑜𝑤 = 𝐻𝑓𝑙𝑜𝑤 ∗ 𝑤 = 0.5𝑚 ∗ 0.005𝑚 = 0.0025𝑚2
The volume flow rate is then calculated by multiplying the flow path cross sectional area
by the average flow velocity.
𝑉̇ = 𝐴𝑓𝑙𝑜𝑤 𝑈𝑓𝑙𝑜𝑤 = 0.0025𝑚2 ∗ 0.002
𝑚
𝑚3
= 0.000005
𝑠
𝑠
To make the units more intuitive, the volume flow rate is converted to liters per minute
𝑉̇ = 0.000005
𝑚3 60𝑠 1000 𝐿 1000 𝑚𝐿
𝐿
∗
∗
∗
= 0.3
3
𝑠 1 𝑚𝑖𝑛 1 𝑚
𝐿
𝑚𝑖𝑛
39
APPENDIX E: TOTAL ENERGY COLLECTED
For a PV/T cell, the Overall efficiency for a condition can be used to obtain the energy
collected from the environment.
𝑄𝑜𝑢𝑡 = 𝑄𝑖 ∗ 𝜂𝑜
The inlet energy is the energy incident from the sun using the dimensions of the PV/T
cell array is calculated below
𝑄𝑖 = 𝑞 ′′ 𝑠𝑢𝑛 𝐴𝐶𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 = 𝑞 ′′ 𝑠𝑢𝑛 ∗ 𝑙 ∗ 𝑤 = 1000
𝑊
∗ 0.5𝑚 ∗ 1.125𝑚 = 562.5 𝑊
𝑚2
The outlet energy of the cell is then taken using the efficiency of the PV/T cell with
many, large fins, in a labyrinth arrangement.
𝑄𝑜𝑢𝑡 = 562.5 𝑊 ∗ 0.7802 = 438.9 𝑊
40
APPENDIX F: CONVECTIVE HEAT LOSS
Convective heat loss for the hybrid solar panel array is given by the below equation.
𝑄̇𝑐𝑜𝑛𝑣 = ℎ ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 ∙ (𝑇𝑎𝑚𝑏 − 𝑇𝑐𝑒𝑙𝑙 )
The heat transfer coefficient of 10.52
𝑊
𝑚2 𝐾
and an ambient temperature of 298.15K are
taken from Table 2. The average PV/T array surface temperature for the labyrinth
arrangement is from Table 7. The collector area is
𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 = 𝑙 ∗ 𝑤 = 0.5𝑚 ∗ 1.125𝑚 = .5625 𝑚2
𝑄̇𝑐𝑜𝑛𝑣 = 10.52
𝑊
∗ 0.5625 𝑚2 (298.15𝐾 − 307.79𝐾)
𝑚2 𝐾
𝑄̇𝑐𝑜𝑛𝑣 = −57.0 𝑊
41
APPENDIX G: RADIATIVE HEAT LOSS
Heat loss to the environment through radiation is described by the Stephan Bolzam
equation.
𝑄̇𝑟𝑎𝑑 = 𝜀 ∗ σ ∗ 𝐴𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 (𝑇𝑎𝑚𝑏 4 − 𝑇𝑐𝑒𝑙𝑙 4 )
The average cell surface temperature
𝑄̇𝑟𝑎𝑑 = 0.60 ∗ 5.67 × 10−8
𝑊
∗ 0.5625 𝑚2 (298.15 𝐾 4 − 307.79𝐾 4 )
𝑚2 𝐾 4
𝑄̇𝑟𝑎𝑑 = −20.5 𝑊
42